Now that we know how to represent a state vector as a superposition of
states, and yet can only measure the state vector to be in one of the
base states. We must determine what happens when we measure the state
vector.

The only way to observe the state of the state vector is to in some
way cause the quantum mechanical system to interact with the
environment. When the state vector is observed it makes a sudden
discontinuous jump to one of the eigenstates. (Williams, Clearwater)

To perform any sort of useful calculation we must be able so say
something about which base state a quantum mechanical system will
collapse into. The probability that the state vector will collapse
into the j'th eigenstate, is given by
| wj|2 which is defined
to be
aj2 + bj2 if
wj = aj + i*bj, where
wj is the complex projection of the state vector onto the j'th
eigenstate. In general the chance of choosing any given state is

Prob(j) =

but
as mentioned earlier we will insist on having the state vector of
length one, and in this case the probability expression simplifies to
Prob(j) = | wj|2.